
where \(\mathcal {C}_{\mathcal {F}}\)  and \(\mathcal {C}_{\mathcal {A}}^\mu \)  are the collision kernels. In the absence of an external field, following the approach as described in Ref. [1]}, we can obtain a Boltzmann-like equation of scalar phase-space distribution functions, \(f^\pm (x,p,s)\)  with an extended phase-space for particles and anti-particles under the relaxation time approximation as,
\(p^\mu \partial _\mu f^\pm (x,\textbf {p},s) = \left(u \cdot p\right) \frac{f^\pm _{\rm eq}(x,\textbf {p},s)-f^\pm (x,\textbf {p},s)}{\tau _{\rm eq}}, \) 
